Limits Are Left-Exact

By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a right-exact functor, but there is a rather huge issue here: the functors are between and , the category of diagrams in while we only defined exactness of functors between categories of modules. The proper way to do this is to introduce the framework of abelian categories and extend our concept of additive functors and exact functors there. However, doing this will take us too far afield so we will prove it directly (which is, admittedly, a bit of a cop out).

Proposition 1.

Let J be an index category, and be diagrams of type J. For concreteness, write these diagrams as

where and . Let be morphisms, written as a collection of over . Then

Note

In summary, taking the limit is left-exact while taking the colimit is right-exact.

Proof

We prove the second claim, leaving the first as an exercise. By proposition 1 here, is concretely described as follows. Take the quotient of by all , where is an arrow in J, and are identified with their images in .

With this description, clearly is surjective. Also, composing is the zero map so . Now write for and for .

Conversely, let represent an element in the kernel of . Thus is a finite sum of . Since is surjective, we can write such a term as

for some . Since is a finite sum of , we can replace x by another representative such that . Then for some . ♦

Neither the limit nor the colimit functor is exact in general. For the colimit case, consider the following commutative diagram of A-modules

where all maps are identities. The rows are short exact sequences and the squares all commute, but taking the colimit of the columns gives

which is not exact.

Exercise A

Find an example for the case of limits.

Direct Limits

We will describe a special case where taking the colimit is exact.

Given a poset , we recall the category whose objects are elements of S, and between any , with equality if and only if . Composition is the obvious one.

Definition.

A poset is called a directed set if for any , there is a such that and .

In other words, a poset is directed if every finite set has an upper bound.

Definition.

If J is an index category obtained from for some directed set S, then a diagram in of type J is called a directed system. The colimit of is called the direct limit and denoted by

.

In other words, direct limit = colimit over directed set. We will abuse notation a little and regard J as the directed set itself.

To avoid set-theoretic difficulties, the directed set J is always assumed to be non-empty.

Example

In exercise C.3 here, for a multiplicative and A-module M, we have an isomorphism of A-modules

where if g is a multiple of f. Since S is multiplicative, any {f, g} has an upper bound fg. Hence is the direct limit of over :

.

Similarly, we have the following direct limit in the category of rings:

.

Next we will discuss the general direct limit in the categories A-Mod and Ring.

Direct Limit of Modules

Let A be a fixed ring; the following holds for direct limits in the category of A-modules.

Proposition 2.

Suppose is a directed system of A-modules over a directed set J. Let

, with canonical for each .

Then for each , there exists an and such that .

Also if satisfies , then there exists such that .

Note

The philosophy is that “whatever happens in the direct limit happens in for some sufficiently large index j“.

Proof

By proposition 1 here, the colimit M is described concretely by taking the quotient of (with canonical ) by relations of the form

Hence any can be written as for . But J is a directed set, so we can pick index such that ; then

, where ,

proving the first claim.

For the second claim, if then is a finite sum of the above relations. Pick an index larger than i and all indices k, l in the sum; then is the sum of the images of these relations in . But each such relation has image in , so as desired. ♦

Corollary 1.

If is a directed system of A-modules such that are all injective, then

is also injective for each .

Finally we have:

Proposition 3.

Let and be directed systems of A-modules and be a morphism of the directed systems, i.e. for any , we have .

If each is injective, so is .

Since taking the colimit is right-exact by proposition 1, we see that taking the direct limit is exact.

Proof

Write and for the canonical maps.

Suppose for . By proposition 2, we have for some ; then

so by proposition 2 again, there exists such that , so . Since is injective we have so . ♦

Exercise B

Describe the direct limit of sets over J. State and prove an analogue of proposition 2.